| L(s) = 1 | + 0.660·2-s − 9.58·3-s − 7.56·4-s − 9.05·5-s − 6.32·6-s − 10.2·8-s + 64.8·9-s − 5.97·10-s − 55.8·11-s + 72.4·12-s − 12.5·13-s + 86.7·15-s + 53.7·16-s + 32.7·17-s + 42.7·18-s + 69.5·19-s + 68.4·20-s − 36.8·22-s − 26.4·23-s + 98.4·24-s − 43.0·25-s − 8.27·26-s − 362.·27-s + 31.4·29-s + 57.2·30-s + 195.·31-s + 117.·32-s + ⋯ |
| L(s) = 1 | + 0.233·2-s − 1.84·3-s − 0.945·4-s − 0.809·5-s − 0.430·6-s − 0.454·8-s + 2.40·9-s − 0.189·10-s − 1.53·11-s + 1.74·12-s − 0.267·13-s + 1.49·15-s + 0.839·16-s + 0.467·17-s + 0.560·18-s + 0.839·19-s + 0.765·20-s − 0.357·22-s − 0.239·23-s + 0.837·24-s − 0.344·25-s − 0.0624·26-s − 2.58·27-s + 0.201·29-s + 0.348·30-s + 1.13·31-s + 0.650·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02243840459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02243840459\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 0.660T + 8T^{2} \) |
| 3 | \( 1 + 9.58T + 27T^{2} \) |
| 5 | \( 1 + 9.05T + 125T^{2} \) |
| 11 | \( 1 + 55.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 362.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 227.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 532.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 20.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 682.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 297.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 804.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 524.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 251.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.426717124734033381805708816606, −7.73851849685303704037792784846, −7.04292524395257017410608466172, −5.98466332793188664518521909790, −5.29099256855965950096615994352, −4.90768787831995783105281559070, −4.09940411475284764756606699111, −3.09135153385854782004386425514, −1.28731780096970470946864009207, −0.079317816809206329690203554639,
0.079317816809206329690203554639, 1.28731780096970470946864009207, 3.09135153385854782004386425514, 4.09940411475284764756606699111, 4.90768787831995783105281559070, 5.29099256855965950096615994352, 5.98466332793188664518521909790, 7.04292524395257017410608466172, 7.73851849685303704037792784846, 8.426717124734033381805708816606
